I have a question about Fisher's equation in a biology context. For example, in Fisher's equation $u_{t} = Du_{xx} + u(1-u)$, where $u$ is a density of cell, the logistic term explains that the capacity is 1.
When we look for a traveling wave solution $U(x-ct)$ (or a heteroclinic orbit), $c$ is a speed, connecting $u=0$ and $u=1,$ do we assume that a traveling wave ansatz must satisfy $0\leq U(x-ct) \leq 1?$ Or, could it be $U(x-ct)>1?$
Thank you so much!
Assuming this is a biological problem and the domain is infinite, then the form of the model that you used indicates a re-scale version, where carrying capacity $u$ is normalized to 1. If the initial profile $u(x,0) \leq 1$ (biological), then $U(x-ct) \leq 1$ for all $t\geq 0$. This is because $u(1-u)\leq 1$ and $Du_{xx}$ ($D>0$) contributes to the diffusion process (e.g. diffusing $u$ from high to lower value across space). If the initial profile $u(x,0) > 1$ (not biological), then it is possible that $U(x-ct) > 1$ for small $t$. However, asymptotically, $U(x-ct)$ will still be bounded by $0$ and $1$, e.g. $0 \leq \liminf U(x-ct) \leq \limsup U(x-ct) \leq 1$.
If the boundary is finite, then the answer depends on the specific boundary conditions and the question needs to be examined case by case.